src/ZF/AC/AC16_WO4.ML

-(*  Title: 	ZF/AC/AC16_WO4.ML
+(*  Title:      ZF/AC/AC16_WO4.ML
     ID:         $Id$
-    Author: 	Krzysztof Grabczewski
+    Author:     Krzysztof Grabczewski
 
   The proof of AC16(n, k) ==> WO4(n-k)
 *)
 
 open AC16_WO4;
 
 (* ********************************************************************** *)
-(* The case of finite set						  *)
+(* The case of finite set                                                 *)
 (* ********************************************************************** *)
 
 goalw thy [Finite_def] "!!A. [| Finite(A); 0<m; m:nat |] ==>  \
-\	EX a f. Ord(a) & domain(f) = a &  \
-\		(UN b<a. f`b) = A & (ALL b<a. f`b lepoll m)";
+\       EX a f. Ord(a) & domain(f) = a &  \
+\               (UN b<a. f`b) = A & (ALL b<a. f`b lepoll m)";
 by (etac bexE 1);
 by (dresolve_tac [eqpoll_sym RS (eqpoll_def RS def_imp_iff RS iffD1)] 1);
 by (etac exE 1);
 by (res_inst_tac [("x","n")] exI 1);
 by (res_inst_tac [("x","lam i:n. {f`i}")] exI 1);
 by (asm_full_simp_tac AC_ss 1);
 by (rewrite_goals_tac [bij_def, surj_def]);
 by (fast_tac (AC_cs addSIs [ltI, nat_into_Ord, lam_funtype RS domain_of_fun,
-	equalityI, singleton_eqpoll_1 RS eqpoll_imp_lepoll RS lepoll_trans,
-	nat_1_lepoll_iff RS iffD2]
-	addSEs [singletonE, apply_type, ltE]) 1);
+        equalityI, singleton_eqpoll_1 RS eqpoll_imp_lepoll RS lepoll_trans,
+        nat_1_lepoll_iff RS iffD2]
+        addSEs [singletonE, apply_type, ltE]) 1);
 val lemma1 = result();
 
 (* ********************************************************************** *)
-(* The case of infinite set						  *)
+(* The case of infinite set                                               *)
 (* ********************************************************************** *)
 
 goal thy "!!x. well_ord(x,r) ==> well_ord({{y,z}. y:x},?s(x,z))";
 by (eresolve_tac [paired_bij RS bij_is_inj RS well_ord_rvimage] 1);
 val well_ord_paired = uresult();

 goal thy "!!A. [| A lepoll B; ~ A lepoll C |] ==> ~ B lepoll C";
 by (fast_tac (FOL_cs addEs [notE, lepoll_trans]) 1);
 val lepoll_trans1 = result();
 
 goalw thy [lepoll_def]
-	"!!X.[| Y lepoll X; well_ord(X, R) |] ==> EX S. well_ord(Y, S)";
+        "!!X.[| Y lepoll X; well_ord(X, R) |] ==> EX S. well_ord(Y, S)";
 by (fast_tac (AC_cs addSEs [well_ord_rvimage]) 1);
 val well_ord_lepoll = result();
 
 goal thy "!!X. [| well_ord(X,R); well_ord(Y,S)  \
-\		|] ==> EX T. well_ord(X Un Y, T)";
+\               |] ==> EX T. well_ord(X Un Y, T)";
 by (eresolve_tac [well_ord_radd RS (Un_lepoll_sum RS well_ord_lepoll)] 1);
 by (assume_tac 1);
 val well_ord_Un = result();
 
 (* ********************************************************************** *)
-(* There exists a well ordered set y such that ...			  *)
+(* There exists a well ordered set y such that ...                        *)
 (* ********************************************************************** *)
 
 goal thy "EX y R. well_ord(y,R) & x Int y = 0 & ~y lepoll z & ~Finite(y)";
 by (res_inst_tac [("x","{{a,x}. a:nat Un  Hartog(z)}")] exI 1);
 by (resolve_tac [Ord_nat RS well_ord_Memrel RS (Ord_Hartog RS
-		well_ord_Memrel RSN (2, well_ord_Un)) RS exE] 1);
+                well_ord_Memrel RSN (2, well_ord_Un)) RS exE] 1);
 by (fast_tac (AC_cs addSIs [Ord_Hartog, well_ord_Memrel, well_ord_paired,
-	equals0I, HartogI RSN (2, lepoll_trans1),
-	subset_imp_lepoll RS (paired_eqpoll RS eqpoll_sym RS
-	eqpoll_imp_lepoll RSN (2, lepoll_trans))]
-	addSEs [RepFunE, nat_not_Finite RS notE] addEs [mem_asym]
-	addSDs [Un_upper1 RS subset_imp_lepoll RS lepoll_Finite,
-	paired_eqpoll RS eqpoll_sym RS eqpoll_imp_lepoll
-	RS lepoll_Finite]) 1);
+        equals0I, HartogI RSN (2, lepoll_trans1),
+        subset_imp_lepoll RS (paired_eqpoll RS eqpoll_sym RS
+        eqpoll_imp_lepoll RSN (2, lepoll_trans))]
+        addSEs [RepFunE, nat_not_Finite RS notE] addEs [mem_asym]
+        addSDs [Un_upper1 RS subset_imp_lepoll RS lepoll_Finite,
+        paired_eqpoll RS eqpoll_sym RS eqpoll_imp_lepoll
+        RS lepoll_Finite]) 1);
 val lemma2 = result();
 
 val [prem] = goal thy "~Finite(B) ==> ~Finite(A Un B)";
 by (fast_tac (AC_cs
-	addSIs [subset_imp_lepoll RS (prem RSN (2, lepoll_infinite))]) 1);
+        addSIs [subset_imp_lepoll RS (prem RSN (2, lepoll_infinite))]) 1);
 val infinite_Un = result();
 
 (* ********************************************************************** *)
-(* There is a v : s_u such that k lepoll x Int y (in our case succ(k))	  *)
-(* The idea of the proof is the following :				  *)
-(* Suppose not, i.e. every element of s_u has exactly k-1 elements of y	  *)
-(* Thence y is less than or equipollent to {v:Pow(x). v eqpoll n#-k}	  *)
-(*   We have obtained this result in two steps :			  *)
-(*   1. y is less than or equipollent to {v:s_u. a <= v}		  *)
-(*      where a is certain k-2 element subset of y			  *)
-(*   2. {v:s_u. a <= v} is less than or equipollent			  *)
-(*      to {v:Pow(x). v eqpoll n-k}					  *)
+(* There is a v : s_u such that k lepoll x Int y (in our case succ(k))    *)
+(* The idea of the proof is the following :                               *)
+(* Suppose not, i.e. every element of s_u has exactly k-1 elements of y   *)
+(* Thence y is less than or equipollent to {v:Pow(x). v eqpoll n#-k}      *)
+(*   We have obtained this result in two steps :                          *)
+(*   1. y is less than or equipollent to {v:s_u. a <= v}                  *)
+(*      where a is certain k-2 element subset of y                        *)
+(*   2. {v:s_u. a <= v} is less than or equipollent                       *)
+(*      to {v:Pow(x). v eqpoll n-k}                                       *)
 (* ********************************************************************** *)
 
 goal thy "!!A. [| ~(EX x:A. f`x=y); f : inj(A, B); y:B |]  \
-\	==> (lam a:succ(A). if(a=A, y, f`a)) : inj(succ(A), B)";
+\       ==> (lam a:succ(A). if(a=A, y, f`a)) : inj(succ(A), B)";
 by (res_inst_tac [("d","%z. if(z=y, A, converse(f)`z)")] lam_injective 1);
 by (fast_tac (AC_cs addSEs [inj_is_fun RS apply_type]
-		addIs [expand_if RS iffD2]) 1);
+                addIs [expand_if RS iffD2]) 1);
 by (REPEAT (split_tac [expand_if] 1));
 by (fast_tac (AC_cs addSEs [left_inverse]) 1);
 val succ_not_lepoll_lemma = result();
 
 goalw thy [lepoll_def, eqpoll_def, bij_def, surj_def]
-	"!!A. [| ~A eqpoll B; A lepoll B |] ==> succ(A) lepoll B";
+        "!!A. [| ~A eqpoll B; A lepoll B |] ==> succ(A) lepoll B";
 by (fast_tac (AC_cs addSEs [succ_not_lepoll_lemma, inj_is_fun]) 1);
 val succ_not_lepoll_imp_eqpoll = result();
 
 val [prem] = goalw thy [s_u_def]
-	"(ALL v:s_u(u, t_n, k, y). k eqpoll v Int y ==> False)  \
-\	==> EX v : s_u(u, t_n, k, y). succ(k) lepoll v Int y";
+        "(ALL v:s_u(u, t_n, k, y). k eqpoll v Int y ==> False)  \
+\       ==> EX v : s_u(u, t_n, k, y). succ(k) lepoll v Int y";
 by (excluded_middle_tac "?P" 1 THEN (assume_tac 2));
 by (resolve_tac [prem RS FalseE] 1);
 by (rtac ballI 1);
 by (etac CollectE 1);
 by (etac conjE 1);
 by (etac swap 1);
 by (fast_tac (AC_cs addSEs [succ_not_lepoll_imp_eqpoll]) 1);
 val suppose_not = result();
 
 (* ********************************************************************** *)
-(* There is a k-2 element subset of y					  *)
+(* There is a k-2 element subset of y                                     *)
 (* ********************************************************************** *)
 
 goalw thy [lepoll_def, eqpoll_def]
-	"!!X. [| n:nat; nat lepoll X |] ==> EX Y. Y<=X & n eqpoll Y";
+        "!!X. [| n:nat; nat lepoll X |] ==> EX Y. Y<=X & n eqpoll Y";
 by (fast_tac (FOL_cs addSDs [Ord_nat RSN (2, OrdmemD) RSN (2, restrict_inj)]
-	addSIs [subset_refl]
-	addSEs [restrict_bij, inj_is_fun RS fun_is_rel RS image_subset]) 1);
+        addSIs [subset_refl]
+        addSEs [restrict_bij, inj_is_fun RS fun_is_rel RS image_subset]) 1);
 val nat_lepoll_imp_ex_eqpoll_n = result();
 
 val ordertype_eqpoll =
-	ordermap_bij RS (exI RS (eqpoll_def RS def_imp_iff RS iffD2));
+        ordermap_bij RS (exI RS (eqpoll_def RS def_imp_iff RS iffD2));
 
 goal thy "!!y. [| well_ord(y,R); ~Finite(y); n:nat  \
-\	|] ==> EX z. z<=y & n eqpoll z";
+\       |] ==> EX z. z<=y & n eqpoll z";
 by (etac nat_lepoll_imp_ex_eqpoll_n 1);
 by (resolve_tac [ordertype_eqpoll RS eqpoll_sym RS eqpoll_imp_lepoll
-	RSN (2, lepoll_trans)] 1 THEN (assume_tac 2));
+        RSN (2, lepoll_trans)] 1 THEN (assume_tac 2));
 by (fast_tac (AC_cs addSIs [nat_le_infinite_Ord RS le_imp_lepoll]
-		addSEs [Ord_ordertype, ordertype_eqpoll RS eqpoll_imp_lepoll
-			RS lepoll_infinite]) 1);
+                addSEs [Ord_ordertype, ordertype_eqpoll RS eqpoll_imp_lepoll
+                        RS lepoll_infinite]) 1);
 val ex_subset_eqpoll_n = result();
 
 goalw thy [lesspoll_def] "!!n. n: nat ==> n lesspoll nat";
 by (fast_tac (AC_cs addSEs [Ord_nat RSN (2, ltI) RS leI RS le_imp_lepoll,
-	eqpoll_sym RS eqpoll_imp_lepoll]
-	addIs [Ord_nat RSN (2, nat_succI RS ltI) RS leI
-	RS le_imp_lepoll RS lepoll_trans RS succ_lepoll_natE]) 1);
+        eqpoll_sym RS eqpoll_imp_lepoll]
+        addIs [Ord_nat RSN (2, nat_succI RS ltI) RS leI
+        RS le_imp_lepoll RS lepoll_trans RS succ_lepoll_natE]) 1);
 val n_lesspoll_nat = result();
 
 goal thy "!!y. [| well_ord(y,R); ~Finite(y); k eqpoll a; a <= y; k: nat |]  \
-\	==> y - a eqpoll y";
+\       ==> y - a eqpoll y";
 by (fast_tac (empty_cs addIs [lepoll_lesspoll_lesspoll]
-	addSIs [Card_cardinal, Diff_lesspoll_eqpoll_Card RS eqpoll_trans,
-		Card_cardinal RS Card_is_Ord RS nat_le_infinite_Ord
-		RS le_imp_lepoll]
-	addSEs [well_ord_cardinal_eqpoll,
-		well_ord_cardinal_eqpoll RS eqpoll_sym,
-		eqpoll_sym RS eqpoll_imp_lepoll,
-		n_lesspoll_nat RS lesspoll_lepoll_lesspoll,
-		well_ord_cardinal_eqpoll RS eqpoll_sym RS eqpoll_imp_lepoll
-		RS lepoll_infinite]) 1);
+        addSIs [Card_cardinal, Diff_lesspoll_eqpoll_Card RS eqpoll_trans,
+                Card_cardinal RS Card_is_Ord RS nat_le_infinite_Ord
+                RS le_imp_lepoll]
+        addSEs [well_ord_cardinal_eqpoll,
+                well_ord_cardinal_eqpoll RS eqpoll_sym,
+                eqpoll_sym RS eqpoll_imp_lepoll,
+                n_lesspoll_nat RS lesspoll_lepoll_lesspoll,
+                well_ord_cardinal_eqpoll RS eqpoll_sym RS eqpoll_imp_lepoll
+                RS lepoll_infinite]) 1);
 val Diff_Finite_eqpoll = result();
 
 goal thy "!!x. [| a<=y; b:y-a; u:x |] ==> cons(b, cons(u, a)) : Pow(x Un y)";
 by (fast_tac AC_cs 1);
 val cons_cons_subset = result();
 
 goal thy "!!x. [| a eqpoll k; a<=y; b:y-a; u:x; x Int y = 0  \
-\	|] ==> cons(b, cons(u, a)) eqpoll succ(succ(k))";
+\       |] ==> cons(b, cons(u, a)) eqpoll succ(succ(k))";
 by (fast_tac (AC_cs addSIs [cons_eqpoll_succ] addEs [equals0D]) 1);
 val cons_cons_eqpoll = result();
 
 goalw thy [s_u_def] "s_u(u, t_n, k, y) <= t_n";
 by (fast_tac AC_cs 1);
 val s_u_subset = result();
 
 goalw thy [s_u_def, succ_def]
-	"!!w. [| w:t_n; cons(b,cons(u,a)) <= w; a <= y; b : y-a; k eqpoll a  \
-\	|] ==> w: s_u(u, t_n, succ(k), y)";
+        "!!w. [| w:t_n; cons(b,cons(u,a)) <= w; a <= y; b : y-a; k eqpoll a  \
+\       |] ==> w: s_u(u, t_n, succ(k), y)";
 by (fast_tac (AC_cs addDs [eqpoll_imp_lepoll RS cons_lepoll_cong]
-		addSEs [subset_imp_lepoll RSN (2, lepoll_trans)]) 1);
+                addSEs [subset_imp_lepoll RSN (2, lepoll_trans)]) 1);
 val s_uI = result();
 
 goalw thy [s_u_def] "!!v. v : s_u(u, t_n, k, y) ==> u : v";
 by (fast_tac AC_cs 1);
 val in_s_u_imp_u_in = result();
 
 goal thy
-	"!!y. [| ALL z:{z: Pow(x Un y) . z eqpoll succ(succ(k))}.  \
-\	EX! w. w:t_n & z <= w; \
-\	k eqpoll a; a <= y; b : y - a; u : x; x Int y = 0 |]  \
-\	==> EX! c. c:{v:s_u(u, t_n, succ(k), y). a <= v} & b:c";
+        "!!y. [| ALL z:{z: Pow(x Un y) . z eqpoll succ(succ(k))}.  \
+\       EX! w. w:t_n & z <= w; \
+\       k eqpoll a; a <= y; b : y - a; u : x; x Int y = 0 |]  \
+\       ==> EX! c. c:{v:s_u(u, t_n, succ(k), y). a <= v} & b:c";
 by (etac ballE 1);
 by (fast_tac (FOL_cs addSIs [CollectI, cons_cons_subset,
-		eqpoll_sym RS cons_cons_eqpoll]) 2);
+                eqpoll_sym RS cons_cons_eqpoll]) 2);
 by (etac ex1E 1);
 by (res_inst_tac [("a","w")] ex1I 1);
 by (rtac conjI 1);
 by (rtac CollectI 1);
 by (fast_tac (FOL_cs addSEs [s_uI]) 1);

 by (assume_tac 2);
 by (fast_tac (AC_cs addSEs [s_u_subset RS subsetD, in_s_u_imp_u_in]) 1);
 val ex1_superset_a = result();
 
 goal thy
-	"!!A. [| succ(k) eqpoll A; k eqpoll B; B <= A; a : A-B; k:nat  \
-\	|] ==> A = cons(a, B)";
+        "!!A. [| succ(k) eqpoll A; k eqpoll B; B <= A; a : A-B; k:nat  \
+\       |] ==> A = cons(a, B)";
 by (rtac equalityI 1);
 by (fast_tac AC_cs 2);
 by (resolve_tac [Diff_eq_0_iff RS iffD1] 1);
 by (rtac equals0I 1);
 by (dresolve_tac [eqpoll_sym RS eqpoll_imp_lepoll] 1);
 by (dresolve_tac [eqpoll_sym RS cons_eqpoll_succ] 1);
 by (fast_tac AC_cs 1);
 by (dtac cons_eqpoll_succ 1);
 by (fast_tac AC_cs 1);
 by (fast_tac (AC_cs addSIs [nat_succI]
-	addSEs [[eqpoll_sym RS eqpoll_imp_lepoll, subset_imp_lepoll] MRS
-	(lepoll_trans RS lepoll_trans) RS succ_lepoll_natE]) 1);
+        addSEs [[eqpoll_sym RS eqpoll_imp_lepoll, subset_imp_lepoll] MRS
+        (lepoll_trans RS lepoll_trans) RS succ_lepoll_natE]) 1);
 val set_eq_cons = result();
 
 goal thy
-	"!!y. [| ALL z:{z: Pow(x Un y) . z eqpoll succ(succ(k))}.  \
-\	EX! w. w:t_n & z <= w; \
-\	ALL v:s_u(u, t_n, succ(k), y). succ(k) eqpoll v Int y;  \
-\	k eqpoll a; a <= y; b : y - a; u : x; x Int y = 0; k:nat   \
-\	|] ==> (THE c. c:{v:s_u(u, t_n, succ(k), y). a <= v} & b:c)  \
-\	Int y = cons(b, a)";
+        "!!y. [| ALL z:{z: Pow(x Un y) . z eqpoll succ(succ(k))}.  \
+\       EX! w. w:t_n & z <= w; \
+\       ALL v:s_u(u, t_n, succ(k), y). succ(k) eqpoll v Int y;  \
+\       k eqpoll a; a <= y; b : y - a; u : x; x Int y = 0; k:nat   \
+\       |] ==> (THE c. c:{v:s_u(u, t_n, succ(k), y). a <= v} & b:c)  \
+\       Int y = cons(b, a)";
 by (dresolve_tac [ex1_superset_a RS theI] 1 THEN REPEAT (assume_tac 1));
 by (rtac set_eq_cons 1);
 by (REPEAT (fast_tac AC_cs 1));
 val the_eq_cons = result();
 

 goal thy "!!a. [| a=b; a=c; b=d |] ==> c=d";
 by (asm_full_simp_tac AC_ss 1);
 val msubst = result();
 
 (* ********************************************************************** *)
-(*   1. y is less than or equipollent to {v:s_u. a <= v}		  *)
-(*      where a is certain k-2 element subset of y			  *)
+(*   1. y is less than or equipollent to {v:s_u. a <= v}                  *)
+(*      where a is certain k-2 element subset of y                        *)
 (* ********************************************************************** *)
 
 goal thy
-	"!!y. [| ALL z:{z: Pow(x Un y) . z eqpoll succ(succ(k))}.  \
-\	EX! w. w:t_n & z <= w; \
-\	ALL v:s_u(u, t_n, succ(k), y). succ(k) eqpoll v Int y;  \
-\	well_ord(y,R); ~Finite(y); k eqpoll a; a <= y;  \
-\	k:nat; u:x; x Int y = 0 |]  \
-\	==> y lepoll {v:s_u(u, t_n, succ(k), y). a <= v}";
+        "!!y. [| ALL z:{z: Pow(x Un y) . z eqpoll succ(succ(k))}.  \
+\       EX! w. w:t_n & z <= w; \
+\       ALL v:s_u(u, t_n, succ(k), y). succ(k) eqpoll v Int y;  \
+\       well_ord(y,R); ~Finite(y); k eqpoll a; a <= y;  \
+\       k:nat; u:x; x Int y = 0 |]  \
+\       ==> y lepoll {v:s_u(u, t_n, succ(k), y). a <= v}";
 by (resolve_tac [Diff_Finite_eqpoll RS eqpoll_sym RS 
-	eqpoll_imp_lepoll RS lepoll_trans] 1
-	THEN REPEAT (assume_tac 1));
+        eqpoll_imp_lepoll RS lepoll_trans] 1
+        THEN REPEAT (assume_tac 1));
 by (res_inst_tac [("f3","lam b:y-a.  \
-\	THE c. c:{v:s_u(u, t_n, succ(k), y). a <= v} & b:c")]
-	(exI RS (lepoll_def RS def_imp_iff RS iffD2)) 1);
+\       THE c. c:{v:s_u(u, t_n, succ(k), y). a <= v} & b:c")]
+        (exI RS (lepoll_def RS def_imp_iff RS iffD2)) 1);
 by (resolve_tac [inj_def RS def_imp_eq RS ssubst] 1);
 by (rtac CollectI 1);
 by (rtac lam_type 1);
 by (resolve_tac [ex1_superset_a RS theI RS conjunct1] 1
-	THEN REPEAT (assume_tac 1));
+        THEN REPEAT (assume_tac 1));
 by (rtac ballI 1);
 by (rtac ballI 1);
 by (resolve_tac [beta RS ssubst] 1 THEN (assume_tac 1));
 by (resolve_tac [beta RS ssubst] 1 THEN (assume_tac 1));
 by (rtac impI 1);
 by (rtac cons_eqE 1);
 by (fast_tac AC_cs 2);
 by (dres_inst_tac [("A","THE c. ?P(c)"), ("C","y")] eq_imp_Int_eq 1);
 by (eresolve_tac [[asm_rl, the_eq_cons, the_eq_cons] MRS msubst] 1
-	THEN REPEAT (assume_tac 1));
+        THEN REPEAT (assume_tac 1));
 val y_lepoll_subset_s_u = result();
 
 (* ********************************************************************** *)
-(* some arithmetic							  *)
+(* some arithmetic                                                        *)
 (* ********************************************************************** *)
 
 goal thy 
-	"!!k. [| k:nat; m:nat |] ==>  \
-\	ALL A B. A eqpoll k #+ m & k lepoll B & B<=A --> A-B lepoll m";
+        "!!k. [| k:nat; m:nat |] ==>  \
+\       ALL A B. A eqpoll k #+ m & k lepoll B & B<=A --> A-B lepoll m";
 by (eres_inst_tac [("n","k")] nat_induct 1);
 by (simp_tac (AC_ss addsimps [add_0]) 1);
 by (fast_tac (AC_cs addIs [eqpoll_imp_lepoll RS
-	(Diff_subset RS subset_imp_lepoll RS lepoll_trans)]) 1);
+        (Diff_subset RS subset_imp_lepoll RS lepoll_trans)]) 1);
 by (REPEAT (resolve_tac [allI,impI] 1));
 by (resolve_tac [succ_lepoll_imp_not_empty RS not_emptyE] 1);
 by (fast_tac AC_cs 1);
 by (eres_inst_tac [("x","A - {xa}")] allE 1);
 by (eres_inst_tac [("x","B - {xa}")] allE 1);

 by (res_inst_tac [("P","%z. z lepoll m")] subst 1 THEN (assume_tac 2));
 by (fast_tac (AC_cs addSIs [equalityI]) 1);
 val eqpoll_sum_imp_Diff_lepoll_lemma = result();
 
 goal thy "!!k. [| A eqpoll succ(k #+ m); B<=A; succ(k) lepoll B;  \
-\	k:nat; m:nat |]  \
-\	==> A-B lepoll m";
+\       k:nat; m:nat |]  \
+\       ==> A-B lepoll m";
 by (dresolve_tac [add_succ RS ssubst] 1);
 by (dresolve_tac [nat_succI RS eqpoll_sum_imp_Diff_lepoll_lemma] 1
-	THEN (REPEAT (assume_tac 1)));
+        THEN (REPEAT (assume_tac 1)));
 by (fast_tac AC_cs 1);
 val eqpoll_sum_imp_Diff_lepoll = result();
 
 (* ********************************************************************** *)
-(* similar properties for eqpoll					  *)
+(* similar properties for eqpoll                                          *)
 (* ********************************************************************** *)
 
 goal thy 
-	"!!k. [| k:nat; m:nat |] ==>  \
-\	ALL A B. A eqpoll k #+ m & k eqpoll B & B<=A --> A-B eqpoll m";
+        "!!k. [| k:nat; m:nat |] ==>  \
+\       ALL A B. A eqpoll k #+ m & k eqpoll B & B<=A --> A-B eqpoll m";
 by (eres_inst_tac [("n","k")] nat_induct 1);
 by (fast_tac (AC_cs addSDs [eqpoll_sym RS eqpoll_imp_lepoll RS lepoll_0_is_0]
-	addss (AC_ss addsimps [add_0])) 1);
+        addss (AC_ss addsimps [add_0])) 1);
 by (REPEAT (resolve_tac [allI,impI] 1));
 by (resolve_tac [succ_lepoll_imp_not_empty RS not_emptyE] 1);
 by (fast_tac (AC_cs addSEs [eqpoll_imp_lepoll]) 1);
 by (eres_inst_tac [("x","A - {xa}")] allE 1);
 by (eres_inst_tac [("x","B - {xa}")] allE 1);
 by (etac impE 1);
 by (fast_tac (AC_cs addSIs [Diff_sing_eqpoll,
-	eqpoll_sym RSN (2, Diff_sing_eqpoll) RS eqpoll_sym]
-	addss (AC_ss addsimps [add_succ])) 1);
+        eqpoll_sym RSN (2, Diff_sing_eqpoll) RS eqpoll_sym]
+        addss (AC_ss addsimps [add_succ])) 1);
 by (res_inst_tac [("P","%z. z eqpoll m")] subst 1 THEN (assume_tac 2));
 by (fast_tac (AC_cs addSIs [equalityI]) 1);
 val eqpoll_sum_imp_Diff_eqpoll_lemma = result();
 
 goal thy "!!k. [| A eqpoll succ(k #+ m); B<=A; succ(k) eqpoll B;  \
-\	k:nat; m:nat |]  \
-\	==> A-B eqpoll m";
+\       k:nat; m:nat |]  \
+\       ==> A-B eqpoll m";
 by (dresolve_tac [add_succ RS ssubst] 1);
 by (dresolve_tac [nat_succI RS eqpoll_sum_imp_Diff_eqpoll_lemma] 1
-	THEN (REPEAT (assume_tac 1)));
+        THEN (REPEAT (assume_tac 1)));
 by (fast_tac AC_cs 1);
 val eqpoll_sum_imp_Diff_eqpoll = result();
 
 (* ********************************************************************** *)
-(* back to the second part						  *)
+(* back to the second part                                                *)
 (* ********************************************************************** *)
 
 goal thy "!!w. [| x Int y = 0; w <= x Un y |]  \
-\	 ==> w Int (x - {u}) = w - cons(u, w Int y)";
+\        ==> w Int (x - {u}) = w - cons(u, w Int y)";
 by (fast_tac (AC_cs addSIs [equalityI] addEs [equals0D]) 1);
 val w_Int_eq_w_Diff = result();
 
 goal thy "!!w. [| w:{v:s_u(u, t_n, succ(l), y). a <= v};  \
-\	l eqpoll a; t_n <= {v:Pow(x Un y). v eqpoll succ(succ(l) #+ m)};  \
-\	m:nat; l:nat; x Int y = 0; u : x;  \
-\	ALL v:s_u(u, t_n, succ(l), y). succ(l) eqpoll v Int y  \
-\	|] ==> w Int (x - {u}) eqpoll m";
+\       l eqpoll a; t_n <= {v:Pow(x Un y). v eqpoll succ(succ(l) #+ m)};  \
+\       m:nat; l:nat; x Int y = 0; u : x;  \
+\       ALL v:s_u(u, t_n, succ(l), y). succ(l) eqpoll v Int y  \
+\       |] ==> w Int (x - {u}) eqpoll m";
 by (etac CollectE 1);
 by (resolve_tac [w_Int_eq_w_Diff RS ssubst] 1 THEN (assume_tac 1));
 by (fast_tac (AC_cs addSDs [s_u_subset RS subsetD]) 1);
 by (fast_tac (AC_cs addEs [equals0D] addSDs [bspec]
-	addDs [s_u_subset RS subsetD]
-	addSEs [eqpoll_sym RS cons_eqpoll_succ RS eqpoll_sym, in_s_u_imp_u_in]
-	addSIs [nat_succI, eqpoll_sum_imp_Diff_eqpoll]) 1);
+        addDs [s_u_subset RS subsetD]
+        addSEs [eqpoll_sym RS cons_eqpoll_succ RS eqpoll_sym, in_s_u_imp_u_in]
+        addSIs [nat_succI, eqpoll_sum_imp_Diff_eqpoll]) 1);
 val w_Int_eqpoll_m = result();
 
 goal thy "!!m. [| 0<m; x eqpoll m; m:nat |] ==> x ~= 0";
 by (fast_tac (empty_cs
-	addSEs [mem_irrefl, ltE, eqpoll_succ_imp_not_empty, natE]) 1);
+        addSEs [mem_irrefl, ltE, eqpoll_succ_imp_not_empty, natE]) 1);
 val eqpoll_m_not_empty = result();
 
 goal thy
-	"!!z. [| z : xa Int (x - {u}); l eqpoll a; a <= y; x Int y = 0; u:x  \
-\	|] ==> cons(z, cons(u, a)) : {v: Pow(x Un y). v eqpoll succ(succ(l))}";
+        "!!z. [| z : xa Int (x - {u}); l eqpoll a; a <= y; x Int y = 0; u:x  \
+\       |] ==> cons(z, cons(u, a)) : {v: Pow(x Un y). v eqpoll succ(succ(l))}";
 by (fast_tac (AC_cs addSIs [cons_eqpoll_succ] addEs [equals0D, eqpoll_sym]) 1);
 val cons_cons_in = result();
 
 (* ********************************************************************** *)
-(*   2. {v:s_u. a <= v} is less than or equipollent			  *)
-(*      to {v:Pow(x). v eqpoll n-k}					  *)
+(*   2. {v:s_u. a <= v} is less than or equipollent                       *)
+(*      to {v:Pow(x). v eqpoll n-k}                                       *)
 (* ********************************************************************** *)
 
 goal thy 
-	"!!x. [| ALL z:{z: Pow(x Un y) . z eqpoll succ(succ(l))}.  \
-\	EX! w. w:t_n & z <= w; \
-\	t_n <= {v:Pow(x Un y). v eqpoll succ(succ(l) #+ m)};  \
-\	0<m; m:nat; l:nat;  \
-\	ALL v:s_u(u, t_n, succ(l), y). succ(l) eqpoll v Int y;  \
-\	a <= y; l eqpoll a; x Int y = 0; u : x  \
-\	|] ==> {v:s_u(u, t_n, succ(l), y). a <= v}  \
-\		lepoll {v:Pow(x). v eqpoll m}";
+        "!!x. [| ALL z:{z: Pow(x Un y) . z eqpoll succ(succ(l))}.  \
+\       EX! w. w:t_n & z <= w; \
+\       t_n <= {v:Pow(x Un y). v eqpoll succ(succ(l) #+ m)};  \
+\       0<m; m:nat; l:nat;  \
+\       ALL v:s_u(u, t_n, succ(l), y). succ(l) eqpoll v Int y;  \
+\       a <= y; l eqpoll a; x Int y = 0; u : x  \
+\       |] ==> {v:s_u(u, t_n, succ(l), y). a <= v}  \
+\               lepoll {v:Pow(x). v eqpoll m}";
 by (res_inst_tac [("f3","lam w:{v:s_u(u, t_n, succ(l), y). a <= v}.  \
-\	w Int (x - {u})")] 
-	(exI RS (lepoll_def RS def_imp_iff RS iffD2)) 1);
+\       w Int (x - {u})")] 
+        (exI RS (lepoll_def RS def_imp_iff RS iffD2)) 1);
 by (resolve_tac [inj_def RS def_imp_eq RS ssubst] 1);
 by (rtac CollectI 1);
 by (rtac lam_type 1);
 by (rtac CollectI 1);
 by (fast_tac AC_cs 1);
 by (rtac w_Int_eqpoll_m 1 THEN REPEAT (assume_tac 1));
 by (simp_tac AC_ss 1);
 by (REPEAT (resolve_tac [ballI, impI] 1));
 by (eresolve_tac [w_Int_eqpoll_m RSN (2, eqpoll_m_not_empty) RS not_emptyE] 1
-	THEN REPEAT (assume_tac 1));
+        THEN REPEAT (assume_tac 1));
 by (dresolve_tac [equalityD1 RS subsetD] 1 THEN (assume_tac 1));
 by (dresolve_tac [cons_cons_in RSN (2, bspec)] 1 THEN REPEAT (assume_tac 1));
 by (etac ex1_two_eq 1);
 by (fast_tac (AC_cs addSEs [s_u_subset RS subsetD, in_s_u_imp_u_in]) 1);
 by (fast_tac (AC_cs addSEs [s_u_subset RS subsetD, in_s_u_imp_u_in]) 1);

 by (fast_tac (empty_cs addSEs [ltE, mem_irrefl]) 1);
 by (fast_tac (empty_cs addSIs [refl, bexI]) 1);
 val ex_eq_succ = result();
 
 goal thy
-	"!!x. [| ALL z:{z: Pow(x Un y) . z eqpoll succ(k)}.  \
-\	EX! w. w:t_n & z <= w; \
-\	well_ord(y,R); ~Finite(y); u:x; x Int y = 0;  \
-\	t_n <= {v:Pow(x Un y). v eqpoll succ(k #+ m)};  \
-\	~ y lepoll {v:Pow(x). v eqpoll m}; 0<k; 0<m; k:nat; m:nat  \
-\	|] ==> EX v : s_u(u, t_n, k, y). succ(k) lepoll v Int y";
+        "!!x. [| ALL z:{z: Pow(x Un y) . z eqpoll succ(k)}.  \
+\       EX! w. w:t_n & z <= w; \
+\       well_ord(y,R); ~Finite(y); u:x; x Int y = 0;  \
+\       t_n <= {v:Pow(x Un y). v eqpoll succ(k #+ m)};  \
+\       ~ y lepoll {v:Pow(x). v eqpoll m}; 0<k; 0<m; k:nat; m:nat  \
+\       |] ==> EX v : s_u(u, t_n, k, y). succ(k) lepoll v Int y";
 by (rtac suppose_not 1);
 by (eresolve_tac [ex_eq_succ RS bexE] 1 THEN (assume_tac 1));
 by (hyp_subst_tac 1);
 by (res_inst_tac [("n1","xa")] (ex_subset_eqpoll_n RS exE) 1
-	THEN REPEAT (assume_tac 1));
+        THEN REPEAT (assume_tac 1));
 by (etac conjE 1);
 by (forward_tac [[y_lepoll_subset_s_u, subset_s_u_lepoll_w] MRS lepoll_trans] 1
-	THEN REPEAT (assume_tac 1));
+        THEN REPEAT (assume_tac 1));
 by (contr_tac 1);
 val exists_proper_in_s_u = result();
 
 (* ********************************************************************** *)
-(* LL can be well ordered						  *)
+(* LL can be well ordered                                                 *)
 (* ********************************************************************** *)
 
 goal thy "{x:Pow(X). x lepoll 0} = {0}";
 by (fast_tac (AC_cs addSDs [lepoll_0_is_0]
-		addSIs [singletonI, lepoll_refl, equalityI]
-		addSEs [singletonE]) 1);
+                addSIs [singletonI, lepoll_refl, equalityI]
+                addSEs [singletonE]) 1);
 val subsets_lepoll_0_eq_unit = result();
 
 goal thy "!!X. [| well_ord(X, R); ~Finite(X); n:nat |]  \
-\		==> EX S. well_ord({Y: Pow(X) . Y eqpoll succ(n)}, S)";
+\               ==> EX S. well_ord({Y: Pow(X) . Y eqpoll succ(n)}, S)";
 by (resolve_tac [well_ord_infinite_subsets_eqpoll_X
-	RS (eqpoll_def RS def_imp_iff RS iffD1) RS exE] 1
+        RS (eqpoll_def RS def_imp_iff RS iffD1) RS exE] 1
     THEN (REPEAT (assume_tac 1)));
 by (fast_tac (ZF_cs addSEs [bij_is_inj RS well_ord_rvimage]) 1);
 val well_ord_subsets_eqpoll_n = result();
 
 goal thy "!!n. n:nat ==> {z:Pow(y). z lepoll succ(n)} =  \
-\	{z:Pow(y). z lepoll n} Un {z:Pow(y). z eqpoll succ(n)}";
+\       {z:Pow(y). z lepoll n} Un {z:Pow(y). z eqpoll succ(n)}";
 by (fast_tac (ZF_cs addIs [le_refl, leI,
-		le_imp_lepoll, equalityI]
-		addSDs [lepoll_succ_disj]
-		addSEs [nat_into_Ord, lepoll_trans, eqpoll_imp_lepoll]) 1);
+                le_imp_lepoll, equalityI]
+                addSDs [lepoll_succ_disj]
+                addSEs [nat_into_Ord, lepoll_trans, eqpoll_imp_lepoll]) 1);
 val subsets_lepoll_succ = result();
 
 goal thy "!!n. n:nat ==>  \
-\	{z:Pow(y). z lepoll n} Int {z:Pow(y). z eqpoll succ(n)} = 0";
+\       {z:Pow(y). z lepoll n} Int {z:Pow(y). z eqpoll succ(n)} = 0";
 by (fast_tac (ZF_cs addSEs [eqpoll_sym RS eqpoll_imp_lepoll 
-		RS lepoll_trans RS succ_lepoll_natE]
-		addSIs [equals0I]) 1);
+                RS lepoll_trans RS succ_lepoll_natE]
+                addSIs [equals0I]) 1);
 val Int_empty = result();
 
 (* ********************************************************************** *)
-(* unit set is well-ordered by the empty relation			  *)
+(* unit set is well-ordered by the empty relation                         *)
 (* ********************************************************************** *)
 
 goalw thy [irrefl_def, trans_on_def, part_ord_def, linear_def, tot_ord_def]
-	"tot_ord({a},0)";
+        "tot_ord({a},0)";
 by (simp_tac ZF_ss 1);
 val tot_ord_unit = result();
 
 goalw thy [wf_on_def, wf_def] "wf[{a}](0)";
 by (fast_tac (ZF_cs addSIs [equalityI]) 1);

 goalw thy [well_ord_def] "well_ord({a},0)";
 by (simp_tac (ZF_ss addsimps [tot_ord_unit, wf_on_unit]) 1);
 val well_ord_unit = result();
 
 (* ********************************************************************** *)
-(* well_ord_subsets_lepoll_n						  *)
+(* well_ord_subsets_lepoll_n                                              *)
 (* ********************************************************************** *)
 
 goal thy "!!y r. [| well_ord(y,r); ~Finite(y); n:nat |] ==>  \
-\	EX R. well_ord({z:Pow(y). z lepoll n}, R)";
+\       EX R. well_ord({z:Pow(y). z lepoll n}, R)";
 by (etac nat_induct 1);
 by (fast_tac (ZF_cs addSIs [well_ord_unit]
-	addss (ZF_ss addsimps [subsets_lepoll_0_eq_unit])) 1);
+        addss (ZF_ss addsimps [subsets_lepoll_0_eq_unit])) 1);
 by (etac exE 1);
 by (eresolve_tac [well_ord_subsets_eqpoll_n RS exE] 1 
-	THEN REPEAT (assume_tac 1));
+        THEN REPEAT (assume_tac 1));
 by (asm_simp_tac (ZF_ss addsimps [subsets_lepoll_succ]) 1);
 by (dtac well_ord_radd 1 THEN (assume_tac 1));
 by (eresolve_tac [Int_empty RS disj_Un_eqpoll_sum RS 
-		(eqpoll_def RS def_imp_iff RS iffD1) RS exE] 1);
+                (eqpoll_def RS def_imp_iff RS iffD1) RS exE] 1);
 by (fast_tac (ZF_cs addSEs [bij_is_inj RS well_ord_rvimage]) 1);
 val well_ord_subsets_lepoll_n = result();
 
 goalw thy [LL_def, MM_def]
-	"!!x. t_n <= {v:Pow(x Un y). v eqpoll n}  \
-\		==> LL(t_n, k, y) <= {z:Pow(y). z lepoll n}";
+        "!!x. t_n <= {v:Pow(x Un y). v eqpoll n}  \
+\               ==> LL(t_n, k, y) <= {z:Pow(y). z lepoll n}";
 by (fast_tac (AC_cs addSEs [RepFunE]
-	addIs [subset_imp_lepoll RS (eqpoll_imp_lepoll
-		RSN (2, lepoll_trans))]) 1);
+        addIs [subset_imp_lepoll RS (eqpoll_imp_lepoll
+                RSN (2, lepoll_trans))]) 1);
 val LL_subset = result();
 
 goal thy "!!x. [| t_n <= {v:Pow(x Un y). v eqpoll n}; \
-\		well_ord(y, R); ~Finite(y); n:nat  \
-\		|] ==> EX S. well_ord(LL(t_n, k, y), S)";
+\               well_ord(y, R); ~Finite(y); n:nat  \
+\               |] ==> EX S. well_ord(LL(t_n, k, y), S)";
 by (fast_tac (FOL_cs addIs [exI]
-		addSEs [LL_subset RSN (2,  well_ord_subset)]
-		addEs [well_ord_subsets_lepoll_n RS exE]) 1);
+                addSEs [LL_subset RSN (2,  well_ord_subset)]
+                addEs [well_ord_subsets_lepoll_n RS exE]) 1);
 val well_ord_LL = result();
 
 (* ********************************************************************** *)
-(* every element of LL is a contained in exactly one element of MM	  *)
+(* every element of LL is a contained in exactly one element of MM        *)
 (* ********************************************************************** *)
 
 goalw thy [MM_def, LL_def]
-	"!!x. [| ALL z:{z: Pow(x Un y) . z eqpoll k}. EX! w. w:t_n & z <= w; \
-\	t_n <= {v:Pow(x Un y). v eqpoll n}; \
-\	v:LL(t_n, k, y)  \
-\	|] ==> EX! w. w:MM(t_n, k, y) & v<=w";
+        "!!x. [| ALL z:{z: Pow(x Un y) . z eqpoll k}. EX! w. w:t_n & z <= w; \
+\       t_n <= {v:Pow(x Un y). v eqpoll n}; \
+\       v:LL(t_n, k, y)  \
+\       |] ==> EX! w. w:MM(t_n, k, y) & v<=w";
 by (step_tac (AC_cs addSEs [RepFunE]) 1);
 by (resolve_tac [lepoll_imp_eqpoll_subset RS exE] 1 THEN (assume_tac 1));
 by (eres_inst_tac [("x","xa")] ballE 1);
 by (fast_tac (AC_cs addSEs [eqpoll_sym]) 2);
 by (res_inst_tac [("a","v")] ex1I 1);

 by (eres_inst_tac [("x","xb")] allE 1);
 by (fast_tac AC_cs 1);
 val unique_superset_in_MM = result();
 
 (* ********************************************************************** *)
-(* The function GG satisfies the conditions of WO4			  *)
-(* ********************************************************************** *)
-
-(* ********************************************************************** *)
-(* The union of appropriate values is the whole x			  *)
+(* The function GG satisfies the conditions of WO4                        *)
+(* ********************************************************************** *)
+
+(* ********************************************************************** *)
+(* The union of appropriate values is the whole x                         *)
 (* ********************************************************************** *)
 
 goal thy
-	"!!x. [| ALL z:{z: Pow(x Un y) . z eqpoll succ(k)}.  \
-\	EX! w. w:t_n & z <= w; \
-\	well_ord(y,R); ~Finite(y); u:x; x Int y = 0;  \
-\	t_n <= {v:Pow(x Un y). v eqpoll succ(k #+ m)};  \
-\	~ y lepoll {v:Pow(x). v eqpoll m}; 0<k; 0<m; k:nat; m:nat  \
-\	|] ==> EX w:MM(t_n, succ(k), y). u:w";
+        "!!x. [| ALL z:{z: Pow(x Un y) . z eqpoll succ(k)}.  \
+\       EX! w. w:t_n & z <= w; \
+\       well_ord(y,R); ~Finite(y); u:x; x Int y = 0;  \
+\       t_n <= {v:Pow(x Un y). v eqpoll succ(k #+ m)};  \
+\       ~ y lepoll {v:Pow(x). v eqpoll m}; 0<k; 0<m; k:nat; m:nat  \
+\       |] ==> EX w:MM(t_n, succ(k), y). u:w";
 by (eresolve_tac [exists_proper_in_s_u RS bexE] 1
-	THEN REPEAT (assume_tac 1));
+        THEN REPEAT (assume_tac 1));
 by (rewrite_goals_tac [MM_def, s_u_def]);
 by (fast_tac AC_cs 1);
 val exists_in_MM = result();
 
 goalw thy [LL_def] "!!w. w : MM(t_n, k, y) ==> w Int y : LL(t_n, k, y)";

 goalw thy [MM_def] "MM(t_n, k, y) <= t_n";
 by (fast_tac AC_cs 1);
 val MM_subset = result();
 
 goal thy 
-	"!!x. [| ALL z:{z: Pow(x Un y) . z eqpoll succ(k)}.  \
-\	EX! w. w:t_n & z <= w; \
-\	well_ord(y,R); ~Finite(y); u:x; x Int y = 0;  \
-\	t_n <= {v:Pow(x Un y). v eqpoll succ(k #+ m)};  \
-\	~ y lepoll {v:Pow(x). v eqpoll m}; 0<k; 0<m; k:nat; m:nat  \
-\	|] ==> EX w:LL(t_n, succ(k), y). u:GG(t_n, succ(k), y)`w";
+        "!!x. [| ALL z:{z: Pow(x Un y) . z eqpoll succ(k)}.  \
+\       EX! w. w:t_n & z <= w; \
+\       well_ord(y,R); ~Finite(y); u:x; x Int y = 0;  \
+\       t_n <= {v:Pow(x Un y). v eqpoll succ(k #+ m)};  \
+\       ~ y lepoll {v:Pow(x). v eqpoll m}; 0<k; 0<m; k:nat; m:nat  \
+\       |] ==> EX w:LL(t_n, succ(k), y). u:GG(t_n, succ(k), y)`w";
 by (forward_tac [exists_in_MM] 1 THEN REPEAT (assume_tac 1));
 by (etac bexE 1);
 by (res_inst_tac [("x","w Int y")] bexI 1);
 by (etac Int_in_LL 2);
 by (rewtac GG_def);
 by (asm_full_simp_tac (AC_ss addsimps [Int_in_LL]) 1);
 by (eresolve_tac [unique_superset_in_MM RS the_equality2 RS ssubst] 1
-	THEN (assume_tac 1));
+        THEN (assume_tac 1));
 by (REPEAT (fast_tac (AC_cs addEs [equals0D] addSEs [Int_in_LL]) 1));
 val exists_in_LL = result();
 
 goalw thy [LL_def] 
-	"!!x. [| ALL z:{z: Pow(x Un y) . z eqpoll k}. EX! w. w:t_n & z <= w; \
-\	t_n <= {v:Pow(x Un y). v eqpoll n};  \
-\	v : LL(t_n, k, y) |]  \
-\	==> v = (THE x. x : MM(t_n, k, y) & v <= x) Int y";
+        "!!x. [| ALL z:{z: Pow(x Un y) . z eqpoll k}. EX! w. w:t_n & z <= w; \
+\       t_n <= {v:Pow(x Un y). v eqpoll n};  \
+\       v : LL(t_n, k, y) |]  \
+\       ==> v = (THE x. x : MM(t_n, k, y) & v <= x) Int y";
 by (fast_tac (AC_cs addSEs [Int_in_LL,
-		unique_superset_in_MM RS the_equality2 RS ssubst]) 1);
+                unique_superset_in_MM RS the_equality2 RS ssubst]) 1);
 val in_LL_eq_Int = result();
 
 goal thy  
-	"!!x. [| ALL z:{z: Pow(x Un y) . z eqpoll k}. EX! w. w:t_n & z <= w; \
-\	t_n <= {v:Pow(x Un y). v eqpoll n};  \
-\	v : LL(t_n, k, y) |]  \
-\	==> (THE x. x : MM(t_n, k, y) & v <= x) <= x Un y";
+        "!!x. [| ALL z:{z: Pow(x Un y) . z eqpoll k}. EX! w. w:t_n & z <= w; \
+\       t_n <= {v:Pow(x Un y). v eqpoll n};  \
+\       v : LL(t_n, k, y) |]  \
+\       ==> (THE x. x : MM(t_n, k, y) & v <= x) <= x Un y";
 by (fast_tac (AC_cs addSDs [unique_superset_in_MM RS theI RS conjunct1 RS 
-	(MM_subset RS subsetD)]) 1);
+        (MM_subset RS subsetD)]) 1);
 val the_in_MM_subset = result();
 
 goalw thy [GG_def] 
-	"!!x. [| ALL z:{z: Pow(x Un y) . z eqpoll k}. EX! w. w:t_n & z <= w; \
-\	t_n <= {v:Pow(x Un y). v eqpoll n};  \
-\	v : LL(t_n, k, y) |]  \
-\	==> GG(t_n, k, y) ` v <= x";
+        "!!x. [| ALL z:{z: Pow(x Un y) . z eqpoll k}. EX! w. w:t_n & z <= w; \
+\       t_n <= {v:Pow(x Un y). v eqpoll n};  \
+\       v : LL(t_n, k, y) |]  \
+\       ==> GG(t_n, k, y) ` v <= x";
 by (asm_full_simp_tac AC_ss 1);
 by (forward_tac [the_in_MM_subset] 1 THEN REPEAT (assume_tac 1));
 by (dtac in_LL_eq_Int 1 THEN REPEAT (assume_tac 1));
 by (rtac subsetI 1);
 by (etac DiffE 1);
 by (etac swap 1);
 by (fast_tac (AC_cs addEs [ssubst]) 1);
 val GG_subset = result();
 
 goal thy  
-	"!!x. [| well_ord(LL(t_n, succ(k), y), S);  \
-\	ALL z:{z: Pow(x Un y) . z eqpoll succ(k)}. EX! w. w:t_n & z <= w; \
-\	well_ord(y,R); ~Finite(y); x Int y = 0;  \
-\	t_n <= {v:Pow(x Un y). v eqpoll succ(k #+ m)};  \
-\	~ y lepoll {v:Pow(x). v eqpoll m}; 0<k; 0<m; k:nat; m:nat  \
-\	|] ==> (UN b<ordertype(LL(t_n, succ(k), y), S).  \
-\	(GG(t_n, succ(k), y)) `  \
-\	(converse(ordermap(LL(t_n, succ(k), y), S)) ` b)) = x";
+        "!!x. [| well_ord(LL(t_n, succ(k), y), S);  \
+\       ALL z:{z: Pow(x Un y) . z eqpoll succ(k)}. EX! w. w:t_n & z <= w; \
+\       well_ord(y,R); ~Finite(y); x Int y = 0;  \
+\       t_n <= {v:Pow(x Un y). v eqpoll succ(k #+ m)};  \
+\       ~ y lepoll {v:Pow(x). v eqpoll m}; 0<k; 0<m; k:nat; m:nat  \
+\       |] ==> (UN b<ordertype(LL(t_n, succ(k), y), S).  \
+\       (GG(t_n, succ(k), y)) `  \
+\       (converse(ordermap(LL(t_n, succ(k), y), S)) ` b)) = x";
 by (rtac equalityI 1);
 by (rtac subsetI 1);
 by (etac OUN_E 1);
 by (eresolve_tac [GG_subset RS subsetD] 1 THEN TRYALL assume_tac);
 by (eresolve_tac [ordermap_bij RS bij_converse_bij RS
-		bij_is_fun RS apply_type] 1);
+                bij_is_fun RS apply_type] 1);
 by (etac ltD 1);
 by (rtac subsetI 1);
 by (eresolve_tac [exists_in_LL RS bexE] 1 THEN REPEAT (assume_tac 1));
 by (rtac OUN_I 1);
 by (resolve_tac [Ord_ordertype RSN (2, ltI)] 1 THEN (assume_tac 2));
 by (eresolve_tac [ordermap_type RS apply_type] 1);
 by (eresolve_tac [ordermap_bij RS bij_is_inj RS left_inverse RS ssubst] 1
-	THEN REPEAT (assume_tac 1));
+        THEN REPEAT (assume_tac 1));
 val OUN_eq_x = result();
 
 (* ********************************************************************** *)
-(* Every element of the family is less than or equipollent to n-k (m)	  *)
+(* Every element of the family is less than or equipollent to n-k (m)     *)
 (* ********************************************************************** *)
 
 goalw thy [MM_def]
-	"!!w. [| w : MM(t_n, k, y); t_n <= {v:Pow(x Un y). v eqpoll n}  \
-\	|] ==> w eqpoll n";
+        "!!w. [| w : MM(t_n, k, y); t_n <= {v:Pow(x Un y). v eqpoll n}  \
+\       |] ==> w eqpoll n";
 by (fast_tac AC_cs 1);
 val in_MM_eqpoll_n = result();
 
 goalw thy [LL_def, MM_def]
-	"!!w. w : LL(t_n, k, y) ==> k lepoll w";
+        "!!w. w : LL(t_n, k, y) ==> k lepoll w";
 by (fast_tac AC_cs 1);
 val in_LL_eqpoll_n = result();
 
 goalw thy [GG_def] 
-	"!!x. [|  \
-\	ALL z:{z: Pow(x Un y) . z eqpoll succ(k)}. EX! w. w:t_n & z <= w; \
-\	t_n <= {v:Pow(x Un y). v eqpoll succ(k #+ m)};  \
-\	well_ord(LL(t_n, succ(k), y), S); k:nat; m:nat |]  \
-\	==> ALL b<ordertype(LL(t_n, succ(k), y), S).  \
-\	(GG(t_n, succ(k), y)) `  \
-\	(converse(ordermap(LL(t_n, succ(k), y), S)) ` b) lepoll m";
+        "!!x. [|  \
+\       ALL z:{z: Pow(x Un y) . z eqpoll succ(k)}. EX! w. w:t_n & z <= w; \
+\       t_n <= {v:Pow(x Un y). v eqpoll succ(k #+ m)};  \
+\       well_ord(LL(t_n, succ(k), y), S); k:nat; m:nat |]  \
+\       ==> ALL b<ordertype(LL(t_n, succ(k), y), S).  \
+\       (GG(t_n, succ(k), y)) `  \
+\       (converse(ordermap(LL(t_n, succ(k), y), S)) ` b) lepoll m";
 by (rtac oallI 1);
 by (asm_full_simp_tac (AC_ss addsimps [ltD,
-	ordermap_bij RS bij_converse_bij RS bij_is_fun RS apply_type]) 1);
+        ordermap_bij RS bij_converse_bij RS bij_is_fun RS apply_type]) 1);
 by (rtac eqpoll_sum_imp_Diff_lepoll 1);
 by (REPEAT (fast_tac (FOL_cs addSDs [ltD]
-	addSIs [eqpoll_sum_imp_Diff_lepoll, in_LL_eqpoll_n]
-	addEs [unique_superset_in_MM RS theI RS conjunct1 RS in_MM_eqpoll_n,
-	in_LL_eq_Int RS equalityD1 RS (Int_lower1 RSN (2, subset_trans)),
-	ordermap_bij RS bij_converse_bij RS bij_is_fun RS apply_type]) 1));
+        addSIs [eqpoll_sum_imp_Diff_lepoll, in_LL_eqpoll_n]
+        addEs [unique_superset_in_MM RS theI RS conjunct1 RS in_MM_eqpoll_n,
+        in_LL_eq_Int RS equalityD1 RS (Int_lower1 RSN (2, subset_trans)),
+        ordermap_bij RS bij_converse_bij RS bij_is_fun RS apply_type]) 1));
 val all_in_lepoll_m = result();
 
 (* ********************************************************************** *)
-(* The main theorem AC16(n, k) ==> WO4(n-k)				  *)
+(* The main theorem AC16(n, k) ==> WO4(n-k)                               *)
 (* ********************************************************************** *)
 
 goalw thy [AC16_def,WO4_def]
-	"!!n k. [| AC16(k #+ m, k); 0 < k; 0 < m; k:nat; m:nat |] ==> WO4(m)";
+        "!!n k. [| AC16(k #+ m, k); 0 < k; 0 < m; k:nat; m:nat |] ==> WO4(m)";
 by (rtac allI 1);
 by (excluded_middle_tac "Finite(A)" 1);
 by (rtac lemma1 2 THEN REPEAT (assume_tac 2));
 by (resolve_tac [lemma2 RS revcut_rl] 1);
 by (REPEAT (eresolve_tac [exE, conjE] 1));

 by (REPEAT (eresolve_tac [exE, conjE] 1));
 by (resolve_tac [well_ord_LL RS exE] 1 THEN REPEAT (assume_tac 1));
 by (fast_tac (AC_cs addSIs [nat_succI, add_type]) 1);
 by (res_inst_tac [("x","ordertype(LL(T, succ(k), y), x)")] exI 1);
 by (res_inst_tac [("x","lam b:ordertype(LL(T, succ(k), y), x).  \
-\	(GG(T, succ(k), y)) `  \
-\	(converse(ordermap(LL(T, succ(k), y), x)) ` b)")] exI 1);
+\       (GG(T, succ(k), y)) `  \
+\       (converse(ordermap(LL(T, succ(k), y), x)) ` b)")] exI 1);
 by (simp_tac AC_ss 1);
 by (fast_tac (empty_cs addSIs [conjI, lam_funtype RS domain_of_fun]
-	addSEs [Ord_ordertype, all_in_lepoll_m, OUN_eq_x]) 1);
+        addSEs [Ord_ordertype, all_in_lepoll_m, OUN_eq_x]) 1);
 qed "AC16_WO4";